Answers

In general, if you subdivide a bezier curve at fixed interval of the default parametrisation, the resulting curve segments will not have the same arc-length. Here is one way to do it http://pomax.github.io/bezierinfo/#tracing.

A while ago, I was playing around with a bit of code (curvature flow) that needed the points to be as uniformly separated as possible. Here is a comparison (without proper labeling on axes! ;)) using linear interpolation and monotone cubic interpolation from the same set of quadrature samples (I used 20 samples per curve, each evaluated using a 24 point gauss-legendre Quadrature) to reparametrise a cubic curve.

[Please note that, this is compared with another run of the algorithm using a lot more nodes and samples taken as ground truth.]